- Cabibbo-Kobayashi-Maskawa matrix
In the

Standard Model ofparticle physics , the**Cabibbo-Kobayashi-Maskawa matrix**(**CKM matrix**,**quark mixing matrix**, sometimes also called**KM matrix**) is aunitary matrix which contains information on the strength of**flavour-changing weak decays**. Technically, it specifies the mismatch ofquantum states ofquark s when they propagate freely and when they take part in theweak interaction s. It is important in the understanding ofCP violation s. A precise mathematical definition of this matrix is given in the article on the formulation of the standard model. This matrix was introduced for three generations of quarks by Makoto Kobayashi andToshihide Maskawa , adding one generation to the matrix previously introduced byNicola Cabibbo . This matrix is also an extension of theGIM mechanism , which only includes 2 of the 3 current families of quarks**The matrix**::$egin\{bmatrix\}\; V\_\{ud\}\; V\_\{us\}\; V\_\{ub\}\; \backslash \; V\_\{cd\}\; V\_\{cs\}\; V\_\{cb\}\; \backslash \; V\_\{td\}\; V\_\{ts\}\; V\_\{tb\}\; end\{bmatrix\}\; egin\{bmatrix\}\; left|\; d\; ight\; angle\; \backslash \; left|\; s\; ight\; angle\; \backslash \; left|\; b\; ight\; angle\; end\{bmatrix\}\; =\; egin\{bmatrix\}\; left|\; d\text{'}\; ight\; angle\; \backslash \; left|\; s\text{'}\; ight\; angle\; \backslash \; left|\; b\text{'}\; ight\; angle\; end\{bmatrix\}$

On the left is the

**CKM Matrix**along with a vector of mass eigenstates of the quarks, and on the right is theweak force eigenstates of the quarks. The CKM matrix describes the probability of a transition from one quark "q" to another quark "q' ". This transition is proportional to $left|\; V\_\{qq\text{'}\}\; ight|\; ^2$.Experimentally, combining a large number of independent measurements, the magnitudes of the values in the matrix have been found to beW.-M. Yao et al., J. Phys. G 33, 1 (2006) and 2007 partial update for the 2008 edition available on the PDG WWW pages (URL: http://pdg.lbl.gov/), [

*http://pdg.lbl.gov/2007/reviews/kmmixrpp.pdf Chapter 11. The CKM Quark-Mixing Matrix*] ] (only central values presented here, uncertainties are excluded):::$V\_\{ij\}\; =\; egin\{bmatrix\}\; 0.97383\; 0.2272\; 0.00396\; \backslash \; 0.2271\; 0.97296\; 0.04221\; \backslash \; 0.00814\; 0.04161\; 0.999100\; end\{bmatrix\}.$

**Counting**To proceed further, it is necessary to count the number of parameters in this matrix,

**V**which appear in experiments, and therefore are physically important. If there are**N**generations of quarks (2**N**flavours) then

#An**N**×**N**complex matrix contains 2**N**^{2}real numbers.

#The constraint of unitarity is ∑_{k}**V**_{ik}**V**^{*}_{jk}= δ_{ij}. Therefore, for the diagonal terms (**i**=**j**) there are**N**constraints, and for the remaining terms,**N**(**N**−1). The number of independent real numbers in a unitary matrix is therefore**N**^{2}.

#One phase can be absorbed into each quark field. An overall common phase is unobservable. Hence there are 2**N**−1 fewer independent numbers, giving the total number of free variables to be (**N**−1)^{2}.

#Of these,**N**(**N**−1)/2 are rotation angles called quark**mixing angles**.

#The remaining (**N**−1)(**N**−2)/2 are complex phases, which causeCP violation .For the case

**N**=2, there is only one parameter which is a mixing angle between two generations of quarks. Historically, this was the first version of CKM matrix when only two generations were known. It is called the**Cabibbo angle**after its inventor Nicola Cabibbo.For the

Standard Model case**N**=3, there are three mixing angles and one CP-violating complex phase.**Observations and predictions**Cabibbo's idea originated from a need to explain two observed phenomena:

#the transitions**u↔d**and**e↔ν**,_{e}**μ↔ν**had similar amplitudes._{μ}

#the transitions with change in strangeness**ΔS=1**had amplitudes equal to 1/4 of those with**ΔS=0**.Cabibbo's solution consisted of postulating weak universality to resolve issue 1, along with a mixing angle**θ**, now called the_{c}**Cabibbo angle**, between the**d**and**s**quarks to resolve issue 2.For two generations of quarks, there are no CP violating phases, as shown by the counting of the previous section. Since CP violations were seen in neutral

kaon decays already in1964 , the emergence of the Standard Model soon after was a clear signal of the existence of a third generation of quarks, as pointed out in1973 by Kobayashi and Maskawa. The discovery of thebottom quark atFermilab (byLeon Lederman 's group) in1976 therefore immediately started off the search for the missing third-generation quark, thetop quark .**Weak universality**The constraints of unitarity of the CKM-matrix on the diagonal terms can be written as::$sum\_k\; |V\_\{ik\}|^2\; =\; 1$

for all generations

**i**. This implies that the sum of all couplings of any of the up-type quarks to all the down-type quarks is the same for all generations. This relation is called**weak universality**afterNicola Cabibbo , who first pointed it out in 1967. Theoretically it is a consequence of the fact that all SU(2) doublets couple with the same strength to thevector boson s of weak interactions. It has been subjected to continuing experimental tests.**The unitarity triangles**The remaining constraints of unitarity of the CKM-matrix can be written in the form::$sum\_k\; V\_\{ik\}V^*\_\{jk\}\; =\; 0.$For any fixed and different

**i**and**j**, this is a constraint on three complex numbers, one for each**k**, which says that these numbers form the sides of a triangle in thecomplex plane . There are six choices of**i**and**j**, and hence six such triangles, each of which is called an**unitary triangle**. Their shapes can be very different, but they all have the same area, which can be related to the CP violating phase. The area vanishes for the specific parameters in the standard model for which there would be no CP violation. The orientation of the triangles depend on the phases of the quark fields.Since the three sides of the triangles are open to direct experiment, as are the three angles, a class of tests of the standard model is to check that the triangle closes. This is the purpose of a modern series of experiments under way at the Japanese BELLE and the Californian

BaBar experiments.**Parameterizations**Four independent parameters are required to fully define the CKM matrix. Many parameterizations have been proposed, and three of the most common ones are shown below.

The original parameterization of Kobayashi and Maskawa used three angles (θ

_{1}, θ_{2}, θ_{3}) and a CP-violating phase (δ). [*M. Kobayashi and T. Maskawa, Progress in Theoretical Physics*] Cosines and sines of the angles are denoted c**49**652 (1973)._{i}and s_{i}, respectively. θ_{1}is the Cabibbo angle.::$egin\{bmatrix\}\; c\_1\; -s\_1\; c\_3\; -s\_1\; s\_3\; \backslash \; s\_1\; c\_2\; c\_1\; c\_2\; c\_3\; -\; s\_2\; s\_3\; e^\{idelta\}\; c\_1\; c\_2\; s\_3\; +\; s\_2\; c\_3\; e^\{idelta\}\backslash \; s\_1\; s\_2\; c\_1\; s\_2\; c\_3\; +\; c\_2\; s\_3\; e^\{idelta\}\; c\_1\; s\_2\; s\_3\; -\; c\_2\; c\_3\; e^\{idelta\}\; end\{bmatrix\}.$

A "standard" parameterization of the CKM matrix uses three Euler angles (θ

_{12}, θ_{23}, θ_{13}) and one CP-violating phase (δ_{13}). [*L. L. Chau and W.-Y. Keung, Physical Review Letters*] Couplings between quark generation i and j vanish if θ**53**1802 (1984)._{ij}= 0. Cosines and sines of the angles are denoted c_{ij}and s_{ij}, respectively. θ_{12}is the Cabibbo angle.::$egin\{bmatrix\}\; c\_\{12\}c\_\{13\}\; s\_\{12\}\; c\_\{13\}\; s\_\{13\}e^\{-idelta\_\{13\; \backslash \; -s\_\{12\}c\_\{23\}\; -\; c\_\{12\}s\_\{23\}s\_\{13\}e^\{idelta\_\{13\; c\_\{12\}c\_\{23\}\; -\; s\_\{12\}s\_\{23\}s\_\{13\}e^\{idelta\_\{13\; s\_\{23\}c\_\{13\}\backslash \; s\_\{12\}s\_\{23\}\; -\; c\_\{12\}c\_\{23\}s\_\{13\}e^\{idelta\_\{13\; -c\_\{12\}s\_\{23\}\; -\; s\_\{12\}c\_\{23\}s\_\{13\}e^\{idelta\_\{13\; c\_\{23\}c\_\{13\}\; end\{bmatrix\}.$

A third parameterization of the CKM matrix was introduced by

Lincoln Wolfenstein with four variables (λ, A, ρ, η) all of order one. [*L. Wolfenstein, Physical Review Letters*] The four Wolfenstein variables are related to the "standard" parameterization:**51**1945 (1983).λ = s

_{12}Aλ

^{2}= s_{23}Aλ

^{3}(ρ-iη) = s_{13}e^{-iδ}The Wolfenstein parameterization of the CKM matrix, to order λ

^{3}, is::$egin\{bmatrix\}\; 1-lambda^2/2\; lambda\; Alambda^3(\; ho-ieta)\; \backslash \; -lambda\; 1-lambda^2/2\; Alambda^2\; \backslash \; Alambda^3(1-\; ho-ieta)\; -Alambda^2\; 1\; end\{bmatrix\}.$

**Nobel Prize**In

2008 , Kobayashi and Maskawa shared one half of theNobel Prize in Physics "for the discovery of the origin of the broken symmetry which predicts the existence of at least three families of quarks in nature". [*cite web| url=http://nobelprize.org/nobel_prizes/physics/laureates/2008/press.html|title=The Nobel Prize in Physics 2008|date=07 October 2008|publisher=nobelprize.org*] Some physicists, especially Italian, had bitter feelings that the Nobel Prize committee failed to reward the work of Cabibbo, on which the work of the other two was based. [*cite web| url=http://www.newscientist.com/article/dn14885-physics-nobel-snubs-key-researcher.html?DCMP=ILC-hmts&nsref=news8_head_dn14885|title=Physics Nobel snubs key researcher|date=07 October 2008*] Asked for a reaction on the prize, Cabibbo preferred to give no comment. According to sources close to him, he was very embittered. [

publisher=New Scientist*cite web| url=http://www.corriere.it/scienze_e_tecnologie/08_ottobre_07/nobel_fisica_italiani_traditi_d9993120-946d-11dd-a0d8-00144f02aabc.shtml|title=Nobel, l'amarezza dei fisici italiani|date=07 October 2008*]

publisher=Corriere della Sera**ee also***Formulation of the standard model and

CP violation s.

*Quantum chromodynamics , flavour and strong CP problem.

*MNS matrix , the equivalent mixing matrix forneutrino s.**Notes****References***cite book | author=Griffiths, David J. | title=Introduction to Elementary Particles | publisher=Wiley, John & Sons, Inc | year=1987 | id=ISBN 0-471-60386-4

*Povh, Bogdan et al., (1995). "Particles and Nuclei: An Introduction to the Physical Concepts". New York: Springer. ISBN 3-540-20168-8**External links***CP violation, by I.I. Bigi and A.I. Sanda (Cambridge University Press, 2000) [ISBN 0-521-44349-0]

* [*http://pdg.lbl.gov/2007/reviews/kmmixrpp.pdf Particle Data Group: the CKM matrix*]

* [*http://pdg.lbl.gov/2007/reviews/cpviolrpp.pdf Particle Data Group: CP violation in meson decays*]

*The [*http://www-public.slac.stanford.edu/babar/ Babar*] experiment atSLAC and the [*http://belle.kek.jp BELLE*] experiment atKEK Japan

* [*http://prola.aps.org/abstract/PRL/v10/i12/p531_1 N. Cabibbo, "Phys. Rev. Lett." 10 (1963) 531.*]

* [*http://www.slac.stanford.edu/spires/find/hep/www?j=PTPKA,49,652 M. Kobayashi and K. Maskawa, "Prog. Theor. Phys." 49 (1973) 652.*]

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2010.*

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